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Higher time derivatives<br />
Higher adiabatic order:<br />
[with S Brahma, E Nelson: arXiv:1208.1242]<br />
¨q = −ω 2 q −U ′ (q)/m<br />
where<br />
− <br />
2m 2 ω U′′′ (q) ( f(q, ˙q)+f 1 (q, ˙q)¨q +f 2 (q)¨q 2 +f 3 (q, ˙q)˙¨q +f 4 (q)¨q )<br />
+···<br />
( ) −1/2 ( ) −5/2<br />
f(q, ˙q) = 1 2<br />
1+ U′′ (q)<br />
mω +<br />
U ′′′′ (q)˙q 2<br />
2 16mω<br />
1+ U′′ (q)<br />
4 mω −<br />
5(U ′′′ (q)) 2 ˙q 2<br />
2 64m 2 ω<br />
(1+ 6<br />
( ) −7/2 ( ) −9/2<br />
− U′′′′′′ (q)˙q 4<br />
64mω<br />
1+ U′′ (q)<br />
6 mω +<br />
21(U ′′′′ (q)) 2 ˙q 4<br />
2 256m 2 ω<br />
1+ U′′ (q)<br />
8 mω 2<br />
( ) −9/2 (<br />
+ 7U′′′′′ (q)U ′′′ (q)˙q 4<br />
64m 2 ω<br />
1+ U′′ (q)<br />
8 mω −<br />
231U ′′′′ (q)(U ′′′ (q)) 2 ˙q 4<br />
2 512m 3 ω<br />
1+ U′′ (q<br />
10 mω 2<br />
( ) −13/2<br />
+ 1155(U′′′ (q)) 4 ˙q 4<br />
4096m 4 ω<br />
1+ U′′ (q)<br />
12 mω 2 CCCC – p. 12